# Variational Inference on the Boolean Hypercube with the Quantum Entropy

arXiv: https://arxiv.org/abs/2411.03759 · HTML: https://mlsanity.com/paper/2411.03759/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 34 references that are in the map.
- First posted on arXiv 6 Nov 2024.

## Builds on (most cited first, 9 of 9)

- [CVXPY: A Python-Embedded Modeling Language for Convex Optimization](https://mlsanity.com/paper/1603.00943/) (2016)
- [Information Theory with Kernel Methods](https://mlsanity.com/paper/2202.08545/) (2022)
- [Solving SDP Faster: A Robust IPM Framework and Efficient Implementation](https://mlsanity.com/paper/2101.08208/) (2021)
- [A Faster Interior-Point Method for Sum-of-Squares Optimization](https://mlsanity.com/paper/2202.08489/) (2022)
- [Mean-field approximation, convex hierarchies, and the optimality of correlation rounding: a unified perspective](https://mlsanity.com/paper/1808.07226/) (2018)
- [Sum-of-Squares Relaxations for Information Theory and Variational Inference](https://mlsanity.com/paper/2206.13285/) (2022)
- [How to calculate partition functions using convex programming hierarchies: provable bounds for variational methods](https://mlsanity.com/paper/1607.03183/) (2016)
- [Statistical and Geometrical properties of regularized Kernel Kullback-Leibler divergence](https://mlsanity.com/paper/2408.16543/) (2024)
- [Degree bounds for Putinar's Positivstellensatz on the hypercube](https://mlsanity.com/paper/2302.12558/) (2023)

## Abstract (arXiv)

In this paper, we derive variational inference upper-bounds on the log-partition function of pairwise Markov random fields on the Boolean hypercube, based on quantum relaxations of the Kullback-Leibler divergence. We then propose an efficient algorithm to compute these bounds based on primal-dual optimization. An improvement of these bounds through the use of ''hierarchies,'' similar to sum-of-squares (SoS) hierarchies is proposed, and we present a greedy algorithm to select among these relaxations. We carry extensive numerical experiments and compare with state-of-the-art methods for this inference problem.